Afgeleide van $$$\sin{\left(x \right)} - \cos{\left(x \right)}$$$

De afgeleide van een som/verschil is de som/het verschil van de afgeleiden:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right) - \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$

De afgeleide van de cosinus is $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:

$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right) = - {\color{red}\left(- \sin{\left(x \right)}\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$

De afgeleide van de sinus is $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$$\sin{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = \sin{\left(x \right)} + {\color{red}\left(\cos{\left(x \right)}\right)}$$

Vereenvoudig:

$$\sin{\left(x \right)} + \cos{\left(x \right)} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$

Dus, $$$\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$$.